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This book is the outcome of a seminar organized by Michael Freedman and Karen Uhlenbeck (the senior author) at the Mathematical Sciences Research Institute in Berkeley during its first few months of existence. Dan Freed (the junior author) was originally appointed as notetaker. The express purpose of the seminar was to go through a proof of Simon Donaldson's Theorem, which had been announced the previous spring. Donaldson proved the nonsmoothability of certain topological four-manifolds; a year earlier Freedman had constructed these manifolds as part of his solution to the four dimensional ; Poincare conjecture. The spectacular application of Donaldson's and Freedman's theorems to the existence of fake 1R4,s made headlines (insofar as mathematics ever makes headlines). Moreover, Donaldson proved his theorem in topology by studying the solution space of equations the Yang-Mills equations which come from ultra-modern physics. The philosophical implications are unavoidable: we mathematicians need physics! The seminar was initially very well attended. Unfortunately, we found after three months that we had covered most of the published material, but had made little real progress towards giving a complete, detailed proof. Mter joint work extending over three cities and 3000 miles, this book now provides such a proof. The seminar bogged down in the hard analysis (56 59), which also takes up most of Donaldson's paper (in less detail). As we proceeded it became clear to us that the techniques in partial differential equations used in the proof differ strikingly from the geometric and topological material.
List of contents
§1 Fake ?4.- Differentiable structures.- Topological 4-manifolds.- Differentiable 4-manifolds.- A surgical failure.- §2 The Yang-Mills Equations.- Connections.- Topological quantum numbers.- The Yang-Mills functional.- Line bundles.- Donaldson's Theorem.- §3 Manifolds of Connections.- Sobolev spaces.- Reducible connections.- A slice theorem.- The parametrized moduli space.- The moduli space.- §4 Cones on ??2.- Slices again.- Structure of the singular point.- Perturbing the metric.- §5 Orientability.- Index bundles.- Components of J.- The element -1.- §6 Introduction to Taubes' Theorem.- Instantons on S4.- A grafting procedure.- Tools from analysis.- Analytic properties of SDYME.- §7 Taubes Theorem.- Blowing up the metric.- The eigenvalue estimate.- The linearized equation.- Taubes' projection.- §8 Compactness.- Compactness and regularity.- Measuring concentrated curvature.- Compactness in ?.- §9 The Collar Theorem.- Decay estimates.- Conformai deformations.- Exponential gauges.- Connectivity of the collar.- §10 The Technique of Fintushel and Stern.- The moduli space for SO(3) bundles.- Reducible connections.- Analytic details.- Appendix A The Group of Sobolev Gauge sTransformations.- Appendix B The Pontrjagin-Thom Construction.- Appendix C Weitzenböck Formulas.- Appendix D The Removability of Singularities.- Appendix E Topological Remarks.
